Ricci Flow on a Class of Noncompact Warped Product Manifolds

Abstract

We consider the Ricci flow on noncompact $$n+1$$ -dimensional manifolds M with symmetries, corresponding to warped product manifolds $$\mathbb {R}\times T^n$$ with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate $$|{{\mathrm{Rm}}}|(p,t) \le C/t$$ for some $$C > 0$$ and all $$p \in M, t \in (1,\infty )$$ holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as $$t \rightarrow \infty $$ ) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.